Servo-Controllers for Linear Time-Invariant Systems with Operational Constraints

A servo-control design method is proposed for multi-input-multi-output linear time-invariant (LTI) systems with box constraints on the control input and output. The proposed control design is based on the Nagumo Theorem, the Comparison Lemma, the min-norm optimal controllers, and it is directly related to the method of Control Barrier Functions. The Nagumo Theorem ensures forward invariance, while the Comparison Lemma is used to derive operational dynamics constraints that can be enforced by the LTI system for any relative degree. The resulting linear dynamics constraints are embedded into a Quadratic Program (QP) formulation to enforce the designated soft operational constraints. This paper shows that an explicit analytical solution to the QP can be obtained using parameter design choices that allow the Karush-Kuhn-Tucker optimality conditions to become decoupled component-wise. The proposed control design yields a continuous piecewise-linear state feedback policy and as a result, the system stability and robustness metrics can be computed using traditional methods.